2018
Том 70
№ 4

All Issues

Kovalev Yu. G.

Articles: 2
Article (Russian)

On the criteria of transversality and disjointness of nonnegative selfadjoint extensions of nonnegative symmetric operators

Kovalev Yu. G.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 495-505

We propose criteria of transversality and disjointness for the Friedrichs and Krein extensions of a nonnegative symmetric operator in terms of the vectors $\{ \varphi j , j \in J\}$ that form a Riesz basis of the defect subspace. The criterion is applied to the Friedrichs and Krein extensions of the minimal Schr¨odinger operator $\scr A$ d with point potentials. We also present a new proof of the fact that the Friedrichs extension of the operator $\scr A$ d is a free Hamiltonian.

Article (Ukrainian)

Point interactions on the line and Riesz bases of δ -functions

Kovalev Yu. G.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1615-1624

We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$. Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf} \bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\prime \prime \bigr\}$. By using this relationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ \delta (x y_j), y_j \in Y \bigr\} $ and their derivatives $\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ but, conversely, form these bases in the case where $d > 0$. We also present the description of the Friedrichs and Krein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative selfadjoint extensions of the operator $A\prime$ are proposed.